Minimal Face Numbers for Volume Rigidity

TL;DR

This paper extends Maxwell's rigidity count from graphs to higher-dimensional simplicial complexes, establishing minimal face numbers for volume rigidity in Euclidean spaces using algebraic and combinatorial tools.

Contribution

It introduces the minimal face counts for volume rigidity of simplicial complexes in any dimension, generalizing Maxwell's graph rigidity results.

Findings

Derived minimal face counts for volume rigidity in all dimensions

Proved properties of the volume rigidity matroid

Established a vertex removal lemma for volume rigidity

Abstract

Maxwell introduced a necessary minimum number of edges in terms of the number of vertices required for a graph to yield a Euclidean rigid generic framework in $\mathbb{R}^3$, this count was generalised to $\mathbb{R}^d$, for all $d\geq 1$. In this paper, we give the analogous minimum number of $k$-simplices, for all $0\leq k\leq d$, required for a pure $d$-dimensional simplicial complex to yield a volume rigid generic framework in $\mathbb{R}^d$, for all $d\geq 1$. In order to do so, we prove some basic facts about the volume rigidity matroid and use exterior algebraic shifting, a recently added tool to the study of volume rigidity. We later prove a volume rigidity Vertex Removal Lemma and use our count to strengthen the statement.